Entropy, Free Energy, and Equilibrium | General Chemistry 3

This chapter explores the concepts of entropy, free energy, and equilibrium, covering: spontaneous processes, entropy and Boltzmann’s equation, entropy changes, standard molar entropy, standard entropy of reaction, the first, second, and third laws of thermodynamics, Gibbs free energy, Gibbs free energy change and spontaneity, standard free energy changes, standard free energies of formation and reaction, free energy and chemical equilibrium.

Spontaneous Processes

Spontaneous Processes:

A spontaneous process is a reaction or change that occurs naturally under certain conditions, without requiring energy input from an external source. These processes generally lead to a decrease in the system's energy, which often makes them exothermic. However, not all spontaneous processes are exothermic.
 

  • Combustion of hydrogen: 2 H2 (g) + O2 (g) → 2 H2O (l)
    This reaction is a spontaneous, with ΔHorxn = - 571.6 kJ.mol-1 < 0 ⇒ exothermic reaction
  • Melting of ice: H2O (s) → H2O (l)
    This process is spontaneous at T > 0°C, even though it is not exothermic, as ΔHofus = +6.0 kJ.mol-1 > 0.

Entropy

Entropy (S):

Entropy is a measure of the disorder or randomness within a system, reflecting the number of possible arrangements (microstates) that particles and energy can adopt. Greater disorder corresponds to higher entropy.
 

Mathematical definition of entropy (Boltzmann’s equation):

Boltzmann’s equation defines entropy as:
 

S = k ln W

S = entropy (in J.K-1)
k = Boltzmann's constant, 1.38 x 10-23 J.K-1
W = number of microstates, or possible arrangements of particles in the system

 

The number of microstates, W, can be determined using:
 

W = XN

W = number of microstates
X = number of possible orientations or configurations for each particle
N = number of particles

 

This formula shows that even small increases in the number of orientations or particles greatly increase the total number of possible microstates, leading to higher entropy.

Entropy Changes

Entropy as a state function:

Entropy is a state function, meaning its value depends only on the initial and final states of the system, not on the path taken. As a result, the change in entropy is calculated as:
 

ΔS = Sfinal - Sinitial

ΔS = entropy change
Sfinal = entropy of the final state
​​​​Sinitial = entropy of the initial state

 

When the randomness of a system increases, ΔS is positive. If randomness decreases, ΔS is negative.

 

Entropy change for an ideal gas:

For an ideal gas expanding or compressing from an initial volume Vi to a final volume Vf, the entropy change is:
 

ΔS = nR ln VfVi

ΔS = entropy change (in J.K-1)
n = number of moles of gas (in mol)
R = ideal gas constant, 8.314 J.mol-1.K-1
Vi and Vf = initial and final volumes (in L or m3)

 

Because the pressure and volume of an ideal gas are inversely proportionnal, the entropy change can also be expressed in terms of initial and final pressures  for a process at constant temperature:
 

ΔS = nR ln PiPf

ΔS = entropy change (in J.K-1)
n = number of moles of gas (in mol)
R = ideal gas constant, 8.314 J.mol-1.K-1
Pi and Pf = initial and final pressures (in atm or Pa)

Standard Entropies

Standard molar entropy (So):

The standard molar entropy is the absolute entropy of one mole of a pure substance in its standard state (1 atm and 298 K). It is expressed in J.K-1.mol-1.

Entropies are absolute values because they are referenced from the zero-point entropy. According to the Third Law of Thermodynamics: The entropy of a perfectly ordered crystalline substance at 0K is zero.
 

Standard entropy of reaction (ΔSorxn):

The standard entropy of reaction can be calculated simply by subtracting the sum of the standard molar entropies of the reactants from the sum of the  standard molar entropies of the products:
 

ΔSorxn = ΣbSo(products) - ΣaSo(reactants)

ΔSorxn = standard entropy of the reaction (in J.K-1.mol-1)
So = standard molar entropy (in J.K-1.mol-1)
a and b = stoichiometric coefficients of the reactants and products

 

For a general reaction: a A + b B → c C + d D:
 

ΔSorxn = [c So(C) + d So(D)] - [a So(A) + b So(B)]
 

 

For the reaction: N2 (g) + 3 H2 (g) ⇌ 2 NH3 (g) 
ΔSorxn = 2 So [NH3] – So [N2] – 3 So [H2]

Entropy Changes in the Universe

The first law of thermodynamics:

The first law of thermodynamics states that energy is conserved; it cannot be created or destroyed, only transferred or transformed. This principle is often expressed as:
 

ΔUuniv = ΔUsys + ΔUsurr = 0

ΔUuniv = change in internal energy of the universe (in J)
ΔUsys = change in internal energy of the system (in J)
ΔUsurr = change in internal energy of the surroundings (in J)

 

The second law of thermodynamics:

The entropy change in the universe is defined as:
 

ΔSuniv = ΔSsys + ΔSsurr

ΔSuniv = entropy change of the universe (in J.K-1)
ΔSsys = entropy change of the system (in J.K-1)
ΔSsurr = entropy change of the surroundings (in J.K-1)

 

The second law of thermodynamics states that, for any spontaneous process, the total entropy of a system and its surroundings increases:
 

ΔSuniv = ΔSsys + ΔSsurr > 0

 

Determining spontaneity with ΔSuniv:

A process’s spontaneity can be determined by evaluating ΔSuniv:

  • If ΔSuniv > 0: The process is spontaneous.
  • If ΔSuniv < 0: The process is non-spontaneous.
  • If ΔSuniv = 0: The process is at equilibrium.
     

Calculating ΔSsurr:

The entropy change in the surroundings can be related to the heat exchange of the system:
 

ΔSsurr = - ∆HsysT   

ΔSsurr = entropy change in the surroundings (in J.K-1)
ΔHsys = enthalpy change of the system (in J)
T = temperature (in K)

 

Free Energy and Reaction Spontaneity

Free energy (G):

Gibbs free energy is a thermodynamic quantity that combines enthalpy (H), entropy (S), and temperature (T) to predict the spontaneity of a reaction under constant pressure and temperature conditions. It is defined by:
 

G = H - TS

G = Gibbs free energy (in J)
H = enthalpy (in J)
T = temperature (in K)
S = entropy (in J.K-1)

 

Gibbs free energy change (ΔG) and spontaneity:

The change in Gibbs free energy during a reaction is calculated by:
 

ΔG = ΔH - TΔS

ΔG = Gibbs free energy change (in J)
ΔH = enthalpy change (in J)
T = temperature (in K)
ΔS = entropy change (in J.K-1)

 

Gibbs criteria for reaction spontaneity:

  • If ΔG < 0: The reaction is spontaneous and will proceed in the forward direction.
  • If ΔG > 0: The reaction is non-spontaneous and will proceed in the reverse direction.
  • If ΔG = 0: The reaction is at equilibrium, with no net change in the concentrations of reactants and products.
     

Temperature dependence of spontaneity:

The spontaneity of a reaction depends on the relative magnitudes of ΔH, T, and ΔS:

  • When ΔH < 0 and ΔS > 0: ΔG is negative at all temperatures, so the reaction is always spontaneous.
  • When ΔH > 0 and ΔS < 0: ΔG is positive at all temperatures, so the reaction is always non-spontaneous.
  • When ΔH < 0 and ΔS < 0: The reaction is spontaneous at low temperatures (when TΔS is small) but becomes non-spontaneous at high temperatures.
  • When ΔH > 0 and ΔS > 0: The reaction is non-spontaneous at low temperatures but becomes spontaneous at high temperatures (when TΔS outweighs ΔH).

Standard Free Energy Changes

Standard-state conditions:

The conditions used by chemists to define the standard states of pure substances and solutions are:

  • Solids, liquids, and gases in pure form at 1 atm pressure.
  • Solutes in solution at a 1 M concentration.
  • Elements in their most stable allotropic form at 1 atm.
  • A specified temperature, typically 298K.

 

Standard free energy change (ΔGo):

The standard free energy change is the Gibbs free energy change for a reaction when all reactants and products are in their standard states at 298K. It can be calculated using standard enthalpy and entropy changes:
 

ΔGo = ΔHo - TΔSo

ΔGo = standard free energy change (in J or kJ)
ΔHo = standard enthalpy change (in J or kJ)
T = temperature (in K)
ΔSo = standard entropy change (in J.K-1)

 

Standard free energy change for the decomposition of calcium carbonate:

CaCO3 ​(s) → CaO(s) + CO2​ (g) with ΔHo = 178.3 kJ and ΔSo = 160.5 J.K-1

ΔGo = ΔHo - TΔSo
ΔGo = 178.3 kJ - (298K)(0.1605 kJ.K-1)
ΔGo = 130.5 kJ
Since ΔGo is positive, this reaction is non-spontaneous under standard conditions.


 

Standard free energy of formation (ΔGfo):

The standard free energy of formation is the change in free energy when 1 mole of a compound forms from its elements in their standard states under standard conditions (298 K and 1 atm). By convention, the ΔGfo of elements in their most stable forms is zero.

 

ΔGfo [O2 (g)] = 0 J.mol-1;  ΔGfo [N2 (g)] = 0 J.mol-1; ΔGfo [C (graphite)] = 0 J.mol-1​​​​​​​.


 

Standard free energy of reaction (ΔGorxn):

The standard free energy of reaction can be calculated simply by subtracting the sum of the standard free energies of formation of the reactants from the sum of the standard free energies of formation of the products:
 

ΔGorxn = Σb ΔGfo(products) - Σa ΔGfo(reactants)

ΔGorxn = standard free energy of the reaction (in J)
ΔGfo = standard free energy of formation (in J.mol-1)
a and b = stoichiometric coefficients of the reactants and products

 

For a general reaction: a A + b B → c C + d D:
 

ΔGorxn = [c ΔGfo(C) + d ΔGfo(D)] - [a ΔGfo(A) + b ΔGfo(B)]

 

 

For the reaction: N2 (g) + 3 H2 (g) ⇌ 2 NH3 (g)
ΔGorxn = 2 ΔGof [NH3] – ΔGof [N2] – 3 ΔG​​​​​​​of [H2]

Free Energy and Chemical Equilibrium

Relationship between ΔG and ΔGo:

The relationship between the free energy change and the standard free energy change is:
 

ΔG = ΔGo + RT ln Q

ΔG = free energy change (in J or kJ)
ΔGo = standard free energy change (in J or kJ)
R = gas constant = 8.314 J.mol-1.K-1
T = temperature (in K)
Q = reaction quotient, representing the ratio of product and reactant concentrations at any point in the reaction.

 

Relationship between ΔGo and K:

When the system reaches equilibrium (Q = K), the free energy change ΔG = 0. Thus, the previous equation simplifies to:
 

ΔGo = - RT ln K

ΔGo = standard free energy change (in J or kJ)
R = gas constant = 8.314 J.mol-1.K-1
T = temperature (in K)
K = equilibrium constant

 

  • If ΔGo < 0 ⇔ ln K > 0 ⇔ K > 1: The reaction favors products.
  • If ΔGo > 0 ⇔ ln K < 0 ⇔ K < 1: The reaction favors reactants.
  • If ΔGo = 0 ⇔ ln K = 0 ⇔ K = 1: Neither products nor reactants are favored.

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Entropy is a measure of the disorder or randomness in a system, denoted by the symbol S. It plays a central role in thermodynamics, describing the number of possible configurations that a system can have. The importance of entropy in chemical processes stems from the Second Law of Thermodynamics, which states that for a spontaneous process, the total entropy of the system and its surroundings must increase.

In chemistry, this translates to the idea that reactions or processes are favorable when they lead to an increase in the overall entropy. Understanding entropy helps chemists predict the spontaneity of chemical reactions, determine energy requirements or yields, and design processes that maximize efficiency.

Several factors can lead to an increase in entropy within a system or process:

  • Temperature Increase: When a system is heated, the energy of its particles increases, leading to more vigorous motion and a greater number of microstates.
  • Physical State Change: Changing from a more ordered phase to a less ordered phase, such as from solid to liquid to gas, increases entropy, because the molecules have more freedom of motion.
  • Dissolution: Dissolving a solid into a solvent increases entropy because the molecules or ions of the solute become more dispersed in the solvent.
  • Chemical Reactions: Reactions where there is an increase in the number of gas molecules or in the number of particles overall generally result in increased entropy.
  • Mixing: Mixing two different substances can increase randomness and disorder, and thus entropy. For instance, when gases mix, the entropy increases due to a greater number of possible distributions of the different gas particles.

Entropy changes for chemical reactions can be represented by the difference between the entropy of the products and the reactants. This is calculated using the formula ΔS = Σ S(products) - Σ S(reactants), where Σ indicates the sum over all species, and S represents the standard molar entropy values of each species.

For phase transitions, the entropy change is associated with the amount of heat absorbed or released (q) at a constant temperature (T) and is given by ΔS = q/T. This relationship is particularly useful for transitions such as melting, boiling, or sublimation where the temperature remains constant as the phase change occurs.

Standard entropies, denoted as S°, are a measure of the absolute entropy of a substance at a pressure of 1 bar and a reference temperature, usually at 298.15 K (25°C). They reflect the amount of disorder or randomness in a substance's structure and are used in calculating the entropy change of a system during chemical reactions. By knowing the standard entropies of reactants and products, one can calculate the total entropy change (ΔS°) for a reaction using the formula ΔS° = ΣS°(products) - ΣS°(reactants). This, along with the enthalpy change (ΔH°), is used to determine the Gibbs free energy change (ΔG°), which predicts the spontaneity of a process at standard conditions.

A decrease in system entropy can be observed when a system undergoes a process that leads to a more ordered state. This typically happens during phase transitions such as freezing, where liquid water, with relatively high entropy due to the movement of molecules, transitions to solid ice, with lower entropy because the molecules are arranged in a fixed, crystalline structure. Additionally, chemical reactions that result in a decrease in the number of gas molecules or formation of complex structures from simpler ones can also lead to a decrease in system entropy. It's important to note that while the system itself may decrease in entropy, the second law of thermodynamics requires that the total entropy of the universe (system plus surroundings) does not decrease for a spontaneous process.

The second law of thermodynamics states that for any spontaneous process, the total entropy of the universe always increases. Entropy, which is a measure of disorder or randomness in a system, tends to increase because there are more ways for energy to be distributed in disordered states than in ordered ones. Therefore, in any natural process, the sum of the entropies of the interacting systems and their surroundings will not decrease.

This law is crucial for determining the direction of spontaneous reactions and for understanding why energy transformations are not 100% efficient. The second law also implies that the universe tends toward a state of maximum entropy, leading to the concept of thermodynamic equilibrium where no further entropy changes occur.

The Gibbs free energy (G) is a thermodynamic quantity that combines enthalpy (H), temperature (T), and entropy (S) into a single value to predict the spontaneity of a process. It is defined by the equation \(G = H - TS\). A spontaneous process, one that occurs without external intervention, is characterized by a decrease in Gibbs free energy (\(\Delta G < 0\)). Conversely, a process with a positive change in Gibbs free energy (\(\Delta G > 0\)) is non-spontaneous and requires energy input to proceed. At equilibrium, the change in Gibbs free energy is zero (\(\Delta G = 0\)), indicating that the system is at its most stable state and no net change will happen without external forces.

The free energy change of a reaction (\( \Delta G \)) is influenced by both temperature and entropy. This relationship is described by the Gibbs free energy equation: \( \Delta G = \Delta H - T\Delta S \), where \( \Delta H \) is the change in enthalpy, \( T \) is the absolute temperature in Kelvin, and \( \Delta S \) is the change in entropy. A higher temperature can increase the \( T\Delta S \) term, potentially making \( \Delta G \) more negative, favoring spontaneity if \( \Delta S \) is positive. Conversely, a negative \( \Delta S \) can make the reaction less favorable at higher temperatures. Thus, entropy can either promote or hinder reaction spontaneity depending on whether the entropy change is positive or negative, respectively.

The relationship between the free energy change (ΔG) and the equilibrium constant (K) for a chemical reaction at a constant temperature is given by the equation ΔG = -RT ln K, where R is the gas constant and T is the temperature in Kelvin. When a reaction is at equilibrium, ΔG is zero, indicating that the system is at its most stable state and no net reaction occurs in either direction. A negative ΔG indicates that the reaction is spontaneous and will proceed in the forward direction until equilibrium is reached, while a positive ΔG suggests that the reaction is non-spontaneous in the forward direction and will not proceed without the input of energy.

The sign of the Gibbs free energy change (∆G) indicates whether a reaction or process is spontaneous under constant temperature and pressure. If ∆G is negative, the process is spontaneous, meaning it can occur without input of additional energy. Conversely, if ∆G is positive, the process is non-spontaneous and requires an input of energy to proceed. If ∆G is zero, the system is at equilibrium, and no net change occurs over time.

The Van't Hoff equation describes how the equilibrium constant for a chemical reaction varies with temperature, providing insight into the reaction's temperature dependence and thermodynamic properties such as enthalpy and entropy changes. The Clapeyron-Clausius equation relates the temperature and pressure dependence of phase transitions, particularly useful for determining the enthalpy change of vaporization or sublimation. Both equations are fundamental in predicting how systems will behave under varying conditions and are instrumental in the fields of physical chemistry and chemical engineering.

The Van’t Hoff equation relates the change in the equilibrium constant, K, to the change in temperature, T, given the standard enthalpy change, ΔH°, for a reaction. It is depicted as: ln(K2/K1) = (-ΔH°/R) * (1/T2 - 1/T1), where K1 and K2 are the equilibrium constants at temperatures T1 and T2 respectively, and R is the gas constant. This equation suggests that if the reaction is exothermic (negative ΔH°), an increase in temperature will result in a decrease in the equilibrium constant, shifting the equilibrium towards the reactants. Conversely, for an endothermic reaction (positive ΔH°), an increase in temperature leads to an increase in the equilibrium constant, shifting the equilibrium towards the products.

The Clapeyron-Clausius equation describes the relationship between the temperature and pressure at which two phases of a substance are in equilibrium, such as solid/liquid or liquid/gas. It shows how the temperature of phase transition changes with pressure and is given by:

\( rac{dP}{dT} = rac{\Delta H_{transition}}{T \Delta V_{transition}} \)

Here, \( rac{dP}{dT} \) is the slope of the phase boundary on a P-T diagram, \( \Delta H_{transition} \) is the enthalpy change of the phase transition, \( T \) is the transition temperature, and \( \Delta V_{transition} \) is the volume change during the transition. It provides a quantitative way to understand how the entropy change (\( \Delta S_{transition} \)) associated with a phase change affects the temperature at which the phase change occurs, since \( \Delta S_{transition} = \Delta H_{transition} / T \).